If the SIM card is properly in it's place, then you should read the following:

Since the iPhone is getting a weak signal and loses the signal within a few seconds, I would say that there is a problem with the antenna area (connectors/antenna).

If full reception is available, but the iPhone loses the signal within 5 to 10 seconds, it would normally be a defective power amplifier or a broken antenna switch (most likely the switch would die first when a phone is dropped).

The "no service" message in those two cases is normal: the phone receives the signal but the BTS (Base Transceiver Station) does not receive a signal from the phone (something like a handshake). So it does not "lock" the "cell" to the phone and goes on; it simply doesn't see or hear the phone, and that is the short explanation for why you get the "no service" message.

But what now ?

If there is nothing to see from the dropping and the phone is still under warranty, go to Apple!

If not, open the phone and check the connector between the dock connector and the logic board: it may be slightly dislocated.

You wrote that you checked the connections. This could mean that the antenna is defective, but check the connections again. Checking is better than buying a new part ;-)

Here is the guide for the dock connector.